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Permutation binomials over finite fields. (English) Zbl 1239.11139
Summary: We prove that if \(x^m + ax^n\) permutes the prime field \( \mathbb{F}_p\), where \( m>n>0\) and \( a\in\mathbb{F}_p^*\), then \( \gcd(m-n,p-1) > \sqrt{p}-1\). Conversely, we prove that if \( q\geq 4\) and \( m>n>0\) are fixed and satisfy \( \gcd(m-n,q-1) > 2q(\log \log q)/\log q\), then there exist permutation binomials over \( \mathbb{F}_q\) of the form \( x^m + ax^n\) if and only if \( \gcd(m,n,q-1) = 1\).

MSC:
11T06 Polynomials over finite fields
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